STOCHASTIC FEATURES 



Random variation about a statistical mean is a characteristic of all growth 

 phenomena. In modeling procedures, the primary objective is to produce estimates of 

 future yields that are the expectations of the overall stand growth process. The 

 approach that is generally used is to assign a random error drawn from an appropriate 

 distribution to each prediction. 



The nature of the distribution of the random component depends intimately upon 

 the resolution of the estimation function with which the random variable is associ- 

 ated. In addition, the self-calibrating feature of this prognosis program influences 

 the distribution of the unexplained variation that is to be represented by the random 

 variable. For example, in the function for diameter change, there are variables that 

 change from tree to tree, other variables change from period to period for the same 

 plot, and a few variables that quantify unchanging characteristics of the stand such 

 as site, elevation, or habitat. Consequently, the unexplained variation about the 

 regression surface will have three components: among trees, among periods, and among 

 stands. The self- calibration procedure serves to remove the "among stand" component 

 of the residual variation, leaving the other two components to be represented by the 

 distribution of the random variable. 



To appreciate the effect of resolution of the estimation function on the distribu- 

 tion of the random variable, consider two functions for diameter changes that differ 

 only by including or excluding a variable that evaluates a significant effect of crown 

 development. Crown development changes slowly with time; therefore, as an explanatory 

 variable it has a high serial correlation from period to period for a single tree. A 

 stand-growth model could use either function (assuming the simulation is not intended 

 to compare pruning alternatives) . However, if the function without the crown develop- 

 ment variable were used, the variance of the random variable would have to be larger, 

 and the serial correlation between the random variables assigned to each tree in 

 successive periods would need to be larger than if the crown variable were included. 

 Either alternative could be used to generate a stand simulation with the same expected 

 values as long as the stand progresses in a way that does not modify the natural 

 correlations between stand density and crovm development. The difference in results 

 between the two alternative formulations would show up only in the variability of 

 repeated runs of the prognosis. The lower the resolution of the components, the 

 greater would be the variability among repeated runs. 



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